I dont know if it is the right title (#explode edges, #detach edges), some times I miss 3ds max when it comes to such tasks, but here you go a simple way to do it in Houdini.
Well all you need to do is add a carve node and the rest is as shown in the picture.
I must add sth, my friend Muhammad K Ahmed and Sandy Wilson La Tourelle mentioned the (Convert Line) sop, well although it convert the line from one single prim to multiple but it doesn't break them apart and they will still have a shared point between each two.